Theorem suplocexprlemlub 8055

Description: Lemma for suplocexpr 8056. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemlub	⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑥,𝐴,𝑦   𝑧,𝐵   𝜑,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑤,𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦<P 𝐵)
2		ltrelpr 7836	. . . . . . . 8 ⊢ <P ⊆ (P × P)
3	2	brel 4807	. . . . . . 7 ⊢ (𝑦<P 𝐵 → (𝑦 ∈ P ∧ 𝐵 ∈ P))
4	3	simpld 112	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝑦 ∈ P)
5	4	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦 ∈ P)
6	3	simprd 114	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝐵 ∈ P)
7	6	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝐵 ∈ P)
8		ltdfpr 7837	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝐵 ∈ P) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
9	5, 7, 8	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
10	1, 9	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))
11		simprrr 542	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (1st ‘𝐵))
12		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
13	12	fveq2i 5678	. . . . . . . . 9 ⊢ (1st ‘𝐵) = (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
14		npex 7804	. . . . . . . . . . . . 13 ⊢ P ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → P ∈ V)
16		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
17		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
18		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
19	16, 17, 18	suplocexprlemss 8046	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
20	15, 19	ssexd 4255	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
21		fo1st 6364	. . . . . . . . . . . . 13 ⊢ 1st :V–onto→V
22		fofun 5596	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
24		funimaexg 5445	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V)
25	23, 24	mpan 424	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (1st “ 𝐴) ∈ V)
26		uniexg 4565	. . . . . . . . . . 11 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V)
27	20, 25, 26	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (1st “ 𝐴) ∈ V)
28		nqex 7694	. . . . . . . . . . 11 ⊢ Q ∈ V
29	28	rabex 4261	. . . . . . . . . 10 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V
30		op1stg 6357	. . . . . . . . . 10 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
31	27, 29, 30	sylancl 413	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
32	13, 31	eqtrid 2279	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐵) = ∪ (1st “ 𝐴))
33	32	eleq2d 2304	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
34	33	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
35	11, 34	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ ∪ (1st “ 𝐴))
36		suplocexprlemell 8044	. . . . 5 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
37	35, 36	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
38		simprl 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ Q)
39	38	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ Q)
40		simprrl 541	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (2nd ‘𝑦))
41	40	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (2nd ‘𝑦))
42		simpr 110	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (1st ‘𝑧))
43		rspe 2593	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
44	39, 41, 42, 43	syl12anc 1272	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
45	4	ad4antlr 495	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦 ∈ P)
46	19	ad4antr 494	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝐴 ⊆ P)
47		simplr 529	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ 𝐴)
48	46, 47	sseldd 3243	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ P)
49		ltdfpr 7837	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
50	45, 48, 49	syl2anc 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
51	44, 50	mpbird 167	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦<P 𝑧)
52	51	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) → (𝑠 ∈ (1st ‘𝑧) → 𝑦<P 𝑧))
53	52	reximdva 2646	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
54	37, 53	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
55	10, 54	rexlimddv 2667	. 2 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
56	55	ex 115	1 ⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526  Vcvv 2815   ⊆ wss 3214  ⟨cop 3697  ∪ cuni 3919  ∩ cint 3954   class class class wbr 4114   “ cima 4757  Fun wfun 5351  –onto→wfo 5355  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611   <_Q cltq 7616  Pcnp 7622  <_P cltp 7626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-qs 6786  df-ni 7635  df-nqqs 7679  df-inp 7797  df-iltp 7801

This theorem is referenced by:  suplocexpr  8056